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Multiscale Interactions in an Idealized Walker Cell: Simulations with Sparse
Space–Time Superparameterization
JOANNA SLAWINSKA
Center for Prototype Climate Modeling, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates
OLIVIER PAULUIS AND ANDREW J. MAJDA
Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University,
New York, New York
WOJCIECH W. GRABOWSKI
National Center for Atmospheric Research, Boulder, Colorado
(Manuscript received 3 March 2014, in final form 2 October 2014)
ABSTRACT
This paper discusses the sparse space–time superparameterization (SSTSP) algorithm and evaluates its
ability to represent interactions between moist convection and the large-scale circulation in the context of
a Walker cell flow over a planetary scale two-dimensional domain. The SSTSP represents convective motions
in each column of the large-scale model by embedding a cloud-resolving model, and relies on a sparse
sampling in both space and time to reduce computational cost of explicit simulation of convective processes.
Simulations are performed varying the spatial compression and/or temporal acceleration, and results are
compared to the cloud-resolving simulation reported previously. The algorithm is able to reproduce a broad
range of circulation features for all temporal accelerations and spatial compressions, but significant biases are
identified. Precipitation tends to be too intense and too localized over warm waters when compared to the
cloud-resolving simulations. It is argued that this is because coherent propagation of organized convective
systems from one large-scale model column to another is difficult when superparameterization is used, as
noted in previous studies. The Walker cell in all simulations exhibits low-frequency variability on a time scale
of about 20 days, characterized by four distinctive stages: suppressed, intensification, active, and weakening.
The SSTSP algorithm captures spatial structure and temporal evolution of the variability. This reinforces the
confidence that SSTSP preserves fundamental interactions between convection and the large-scale flow, and
offers a computationally efficient alternative to traditional convective parameterizations.
1. Introduction
The interplay between moist convection and the largescale flow is the fundamental feature of the tropical atmosphere. However, the extreme range of spatial and
temporal scales involved makes it difficult to resolve all
relevant processes in numerical models. In large-scale
models, this issue has traditionally been addressed
through the use of convective parameterizations that
account for effects of convective motions on the mean
Corresponding author address: Joanna Slawinska, Center for Prototype Climate Modeling, New York University Abu Dhabi, P.O.
Box 129188, Saadiyat Island, Abu Dhabi, United Arab Emirates.
E-mail: [email protected]
DOI: 10.1175/MWR-D-14-00082.1
Ó 2015 American Meteorological Society
atmospheric temperature and humidity profiles. It is well
recognized, however, that convective parameterizations
fail to reproduce many important features of the tropical
atmosphere. This is partly because many aspects of convection, such as downdrafts, cold pools, and mesoscale
organization, are either excluded or poorly represented in
the parameterizations. Moreover, the parameterizations
often do not reproduce the intrinsic intermittency of
moist convection. This motivates the development of new
approaches to improve the representation of convection
in multiscale simulations of the tropical atmosphere.
One way to improve such simulations is to take
advantage of cloud-resolving modeling. Cloud models
emerged in the 1970s (e.g., Steiner 1973; Schlesinger 1975;
Klemp and Wilhelmson 1978; Clark 1979) to study
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MONTHLY WEATHER REVIEW
individual clouds in short simulations (tens of minutes)
and typically applied idealized forcing techniques (e.g.,
initiating cloud development via a warm bubble). More
recently such models have been used in significantly longer simulations (days and weeks) and apply large computational domains. Such simulations are often driven by
observationally based time-evolving large-scale forcings
and thus allow for better comparison with observations
(e.g., Grabowski et al. 1996, 1998; Xu and Randall 1996;
Xu et al. 2002; Fridlind et al. 2012, among many others).
Cloud-resolving models solve nonhydrostatic governing
equations and allow convective development in conditionally unstable conditions. The horizontal resolution of
;1 km is high enough for the simulation of the dynamical
evolution of individual clouds, with microphysical, turbulent, and radiative processes required to be parameterized.
Explicit representation of cloud dynamics allows capturing key features that convective parameterizations struggle with. During the last 30 years, many studies focused on
statistical response of cloud ensembles to the large-scale
forcing over a limited area (e.g., Soong and Ogura 1980;
Soong and Tao 1980; Tao and Soong 1986). So far, cloudresolving models appear superior to any kind of convective parameterization, as found by comparing model
results to observations. However, the computational cost
still severely limits global cloud-resolving simulations and
alternative approaches need to be explored. An important
application of cloud-resolving modeling in the context of
global simulation is validation and improvement of other
approaches designed to estimate feedbacks from convective to mesoscale, synoptic, and global scales.
Rescaling approaches have also been suggested to
extend cloud-resolving modeling to global simulations.
The underlying idea is to artificially reduce the scale
separation between convective and planetary scales, and
thus to make explicit simulation of convection computationally feasible in global domains. The Diabatic Acceleration and Rescaling (DARE) approach (Kuang
et al. 2005) and the hypo-hydrostatic approach (Pauluis
et al. 2006; Garner et al. 2007) are examples of such
techniques. In DARE, Earth’s diameter is reduced, the
rotation rate is increased, and diabatic processes are
accelerated. In the hypo-hydrostatic approach, the vertical acceleration is rescaled. Pauluis et al. (2006) have
shown that both approaches are mathematically equivalent and they reduce the scale separation between
convection and the planetary scale without affecting the
dynamics at large scales. However, changes in the behavior of convection due to the rescaling limit the applicability of these methods. Nevertheless, they illustrate
how mathematical rescaling can offer a computationally
efficient way to use cloud-resolving models in global
simulations.
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The second approach is to take advantage of a cloudresolving model for global simulations through the
superparameterization methodology (Grabowski and
Smolarkiewicz 1999; Grabowski 2001, 2004; Randall et al.
2003; Majda and Grooms 2014). In this framework, a twodimensional cloud-resolving model with periodic lateral
boundaries is embedded within each column of a global
model to simulate interactions between convective and
global scales. The simulated large-scale flow includes the
convective feedback from small to large scales, and convective scales respond to the forcing from the large-scale
dynamics. In the original superparameterization (SP),
convective feedback is calculated using a cloud-resolving
model applying horizontal domain equal (or approximately equal) to the large-scale model horizontal grid
length. Grabowski (2001, section 3) simulated a 2D
Walker cell of 4000-km horizontal extent applying the SP
approach with different horizontal grid lengths (from 20
to 500 km) and thus different extents of the SP model
horizontal domain. Results from these simulations were
compared to the fully resolved simulations [described in
Grabowski et al. (2000)] as well as between each other. SP
seemed reasonably successful in reproducing large-scale
conditions as simulated by the cloud-resolving model
(e.g., dry subsidence and humid ascent regions, largescale flow featuring the first and second baroclinic modes,
etc.). However, mesoscale organization of convection
and the strength of the quasi-two-day oscillations, the
prominent feature of the fully resolved simulations, were
significantly different between SP simulations.
Over the last 10 years, SP has been tested in many
studies of tropical dynamics. Khairoutdinov et al. (2005)
and DeMott et al. (2007) found that while Madden–Julian
oscillation (MJO) is missing from the standard Community Atmosphere Model (CAM), it is simulated reasonably well with SP-CAM (i.e., the superparameterized
CAM). They report several important improvements in
simulating tropical climatology, such as a more realistic
distribution of cirrus cloudiness or intense precipitation.
However, some important biases persist, for instance, too
heavy precipitation over the western tropical Pacific associated with the Indian monsoon or too low shallowconvection cloud fraction and light rain across parts of the
tropics and subtropics. Studies attempting to explain the
reason of excessive precipitation in the western Pacific
during monsoon periods typically find significant correlation between moisture content in the column and precipitation. Thayer-Calder and Randall (2009) suggest that
the difference comes from contrasting profiles of convective heating that excite different large-scale circulation
(and thus affect surface wind and evaporative feedback)
and subsequently differently moisten the troposphere. Luo
and Stephens (2006) argue that convection–evaporation
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SLAWINSKA ET AL.
feedback is the main culprit of excessive rain and suggest
that this may be due to the periodicity of SP’s cloudresolving models leading to the prolonged presence of
precipitating convection at a given location.
The mathematical aspects of the SP implementation
are important as illustrated by the above examples and
other studies (e.g., Grabowski 2004). Over the years,
several algorithms have been proposed to implement
the SP framework. Here, we evaluate the ability of the
sparse space–time superparameterization (SSTSP) to
accurately reproduce the interactions between convection and the large-scale flow. In the SSTSP framework,
described in more detail in the next section, the embedded cloud-resolving model uses a horizontal domain
that is small in comparison to the horizontal grid length
of the large-scale model, and for a time period that is
short when compared to the time step of the large-scale
model. Here we apply SSTSP to perform systematic
tests of impact of spatial compression applied sometimes
in the previous SP implementations (Xing et al. 2009;
Wang et al. 2011; Pritchard et al. 2011; Kooperman et al.
2013) and temporal acceleration. The latter one is similar to the DARE and hypo-hydrostatic rescaling, thus
significantly increasing computational efficiency of the
approach. As with the original SP, the goal of the SSTSP
algorithm is to obtain a statistically correct representation of the convective impact on the large-scale flow at
a reduced computational cost.
Preliminary SSTSP testing reported in Xing et al.
(2009) applied two-dimensional simulations of an idealized squall line propagating in a periodic horizontal
domain of 1024 km. The performance of the SSTSP algorithm was examined for a range of environmental
conditions that differed in the prescribed vertical shear
of the large-scale horizontal wind. The SSTSP algorithm
seemed to capture propagation of the squall line and its
speed. In particular, propagation speed appeared to be
strongly controlled by the vertical profile of the largescale shear, with no significant drawbacks of the SSTSP
algorithm. Contrasting convective organizations were
simulated for different shears, from the squall line to decaying convection. This provided hope for the SSTSP algorithm in simulations of different convective regimes for
various large-scale conditions. Furthermore, structural
agreement was found for large-scale features of simulated
convective systems since pattern correlation was high for
horizontal velocity or specific humidity. However, the
impact of the SSTSP algorithm on large-scale features
(e.g., the mean temperature and moisture profiles) was
severely limited because of the short simulation time
(36 h) and relatively small computational domain.
Here, we investigate the accuracy of the SSTSP algorithm in reproducing interactions between convection
565
and the large-scale flow in an idealized Walker cell circulation. We compare SSTSP results against the benchmark solution obtained with the cloud-resolving model.
The latter is described in more detail in Slawinska et al.
(2014, hereafter SPMG) focusing on the intraseasonal
variability of the Walker cell with the time scale of
about 20 days. The low-frequency oscillation features
four phases: the suppressed, intensification, active, and
weakening. Intensification of the circulation is associated with the broadening of the large-scale ascent region, which in turn is strongly coupled to propagating
synoptic-scale systems. Details of the SSTSP framework and its implementation are given in section 2.
Results of simulations applying the SSTSP framework
are discussed in section 3 and are compared to the results from SPMG’s cloud-resolving model. Section 4
provides a discussion of the model results and concludes the paper.
2. Model and experimental setup
In this study, we use the anelastic nonhydrostatic atmospheric model EULAG [Smolarkiewicz and Margolin
1997; see Prusa et al. (2008) for a comprehensive review].
EULAG applies finite-difference dynamics based on
the multidimensional positive definite advection transport algorithm (MPDATA) scheme (Smolarkiewicz
2006), which is monotonic and intrinsically dissipative.
The model is used without any additional subgrid-scale
diffusion and a gravity wave absorber is added in the
uppermost 8 km of the domain. The one-moment microphysical scheme of Grabowski (1998) that solves
equations for cloud water and precipitation is applied
here. These two classes correspond to either liquid
(cloud water and rain) or solid (cloud ice and snow) condensate, for temperatures above 268 K or below 253 K,
respectively, and mix of these two for the temperature
range in between. Notably, condensation is calculated with
no supersaturation allowed as defined with respect to either water or ice or a mix of the two, depending on the
temperature. Also autoconversion, accretion, evaporation,
and fallout of precipitation are taken into account.
Here, we apply the SP methodology (Grabowski 2001,
2004) and implement the SSTSP framework as described briefly below [see also Xing et al. (2009)].
a. SP and SSTSP frameworks
1) LARGE-SCALE AND CLOUD-RESOLVING MODEL
EQUATIONS
The large-scale and cloud-resolving models calculate
the evolution of the large-scale F and small-scale u
variables:
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F 5 [U, W, Q, Qy , Qc , Qp ], and
(1)
u 5 [u, w, u, qy , qc , qp ] .
(2)
The variables are the horizontal (U and u) and vertical
(W and w) velocities, potential temperature (Q and u),
water vapor (Qy and qy), condensed water/ice (Qc and
qc), and precipitating water/ice (Qp and qp) mixing ratios, the latter two following the representation of moist
thermodynamics of Grabowski (1998). Evolution of F
and u can be symbolically written as
›F
1 AF 5 SF 1 F CS
F ,
›t
›u
1 Au 5 Su 1 F LS
u ,
›t
and
(3)
(4)
where AF [ (1/ro )(›/›Xj )(ro Uj F) and Au [(1/ro )(›/›xj )
(ro uj u) represent the large-scale and small-scale advection terms, respectively; and SF and Su represent various source terms in the large-scale and small-scale
models, respectively (such as the buoyancy, pressure
gradient, radiative cooling, surface fluxes, phase changes
of the water substance, precipitation formation and fallout, gravity wave absorber, etc.). Surface fluxes and radiative transfer are calculated here in the large-scale
model, while phase changes and precipitation formation/
fallout are considered in the small-scale model only and
they affect the large-scale fields through the small-scale
feedback. In general, one needs to ensure that a given
source is included only once between the two models,
that is, no double counting takes place. The source terms
SF and Su need to be appropriately designed between the
two models. For instance, the pressure gradient terms are
independently formulated between the models (e.g., via
the anelastic continuity equation). The variable FFCS is the
small-scale feedback, and FuLS stands for the large-scale
forcing. The latter two terms represent the coupling between the two models and in general equal the difference
in between large- and small-scale soundings as calculated
by large-scale and embedded cloud-resolving model. A
more detailed explanation is given in sections 2a(2) and
2a(3). The horizontally averaged vertical velocity at
each level of the small-scale model has to vanish because
of the periodic lateral boundary conditions, and the
vertical velocity field cannot be coupled between the
two models.
2) COUPLING PROCEDURE IN SP
The original implementation of the SP is as follows
(cf. Grabowski 2004). Every large-scale grid, DX 3 DZ,
contains a cloud-resolving (small scale) model that has
Nx 3 Nz grid points of grid size Dx 3 Dz, for which
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DX 5 Nx Dx;
DZ 5 Dz,
(5)
that is, the horizontal extent of the small-scale model
domain is equal to the horizontal grid length of the
large-scale model, and the two models share the same
vertical grid. For a given large-scale time step DT, the
evolution from time T to T 1 DT of the large-scale
variable is calculated first:
T1DT
Fj
T1DT
T
5 Fj 1 DT(AF 1 SF )jT
T
CS
1 DTFF
j ,
(6)
standing for the transport and
with DT(AF 1 SF )jT1DT
T
large-scale sources over the period (T: T 1 DT) and
FFCS jT representing the small-scale feedback calculated
at previous time T, as given by (10).
With the large-scale variables already known at (T 1
DT), the vertical profiles of large-scale forcing for the
small-scale variables, FuLS jT , are formulated as follows:
FuLS j 5
T
T1DT
Fj
T
Nx
2 , uj . j1
,
DT
(7)
x
where hijN
1 stands for the horizontal averaging over the
Nx points of the small-scale model. With the large-scale
forcing formulated as above and assumed constant for
the large-scale time step, the small-scale model equations are advanced from T to T 1 DT:
T1DT
uj
T1iDt
5 uj 1 å Dt(Au 1 Su )
i51
T
Nt
T1(i21)Dt
T
1 å DtFuLS ,
i51
Nt
(8)
over Nt time steps for which
DT 5 Nt Dt .
(9)
Finally, at the end of the large-scale model time step,
average profiles of the small-scale feedback are formulated as
N
CS T1DT
FF
j
5
hujT1DT ij1 x 2 FjT1DT
.
DT
(10)
Repeating (6), (7), (8), and (10) allows stepping forward
in time of the combined small-scale and large-scale
system.
3) SPARSE SPACE–TIME ALGORITHM
The sparse space–time algorithm (Xing et al. 2009)
reduces the computational cost of the SP by decreasing
the horizontal extent of the small-scale domain by
a factor of px (i.e., px smaller number of model columns;
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SLAWINSKA ET AL.
reduced space strategy) and the number of small-scale
time steps by a factor of pt (reduced time strategy). In
such a case, the number of small-scale time steps in every large-scale time step and the number of columns in
every large-scale grid, Npt and Npx , are given by
Np 5
Nt
, and
pt
(11)
Np 5
Nx
.
px
(12)
t
x
As in the original SP, the evolution of the large-scale
variables is calculated first according to (6). Then, profiles
of the large-scale forcings are calculated for the smallscale domain of Npx horizontal columns similarly to (7):
Np
FuLS jT
FjT1DT 2 hujT ij1 x
,
5 pt
DT
(13)
but including pt (i.e., adding the time rescaling of the largescale forcing), and applying horizontal averaging over the
N
rescaled small-scale domain (marked hij1 px ). Subsequently,
accelerated evolution of the small-scale variables over Npt
time steps is calculated similarly to (8):
ujT1ðDT =pt Þ
T1iDt
5 uj 1 å Dt(Au 1 Su )
i51
Np
T
t
Np
t
1å
i51
T1(i21)Dt
T
DtFuLS .
(14)
The small-scale variables at the end of the large-scale
time step, ujT1DT, are assumed equal to the solution of
(14) with accelerated forcing:
T1DT
uj
T1(DT/pt)
5 uj
.
(15)
Finally, profiles of the small-scale feedback are computed as
FuCS jT1DT 5
ujT1(DT/ pt) 2 FjT1DT
.
DT
(16)
Elementary considerations [similar to those involving
Eqs. (10) and (11) in Grabowski (2004)] document that
the SSTSP algorithm outlined above ensures appropriate transfer of information between the small-scale and
large-scale models despite spatial compression and
temporal acceleration. For instance, if either SF in (3) or
Su in (4) is assumed constant, then the tendency due to
this source is correctly passed from one model to another (i.e., from the large-scale to small-scale model for
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SF and vice versa for Su) when spatial compression and
temporal acceleration are applied.
Beyond mathematical consistency, one should be also
aware of physical limitations of the SSTSP methodology. For the spatial compression, a small horizontal
domain of the small-scale model may affect not only the
statistical sampling of small-scale features, but their
evolution as well, evolution of convective cells in particular. Since the mean vertical velocity within SP
models at any level has to vanish (because of periodic
lateral boundary conditions), the upward convective
mass flux has to be balanced by the environmental subsidence. The key point is that the vertical development of
convective clouds may be affected when the computational domain is reduced to a small number of columns.
The temporal acceleration is perhaps more difficult to
interpret. The approach taken in Xing et al. (2009) and
followed here [cf. (16)] implies that the large-scale
forcing is increased in proportion to the temporal
acceleration factor pt. The idea is that the original largescale forcing has to be increased so the small-scale processes can appropriately respond over the pt-shorter time.
An alternative approach might be to keep the large-scale
forcing unchanged, but instead increase the small-scale
feedback by pt. In other words, the small-scale response
to the original feedback would be calculated only for the
pt fraction of DT and linearly extrapolated (i.e., increased
by a factor of pt) before being applied to the large-scale
model. Such a procedure would lead to the same evolution in time of C and u for the case of a constant source.
However, considering fundamental differences between
time scales involved in small-scale and large-scale processes, extrapolation of the small-scale response seems
more problematic than scaling up the large-scale forcing.
b. Experimental design
The developments presented in the previous section are
tested applying the Walker cell circulation in the twodimensional domain following SPMG. As in SPMG,
the environmental profiles come from a simulation of
radiative-convective equilibrium applying a cloud-resolving
model, the System for Atmospheric Modeling, with the
National Center for Atmospheric Research (NCAR)
CAM3 interactive radiation scheme (Khairoutdinov and
Randall 2003). The planetary-scale circulation is driven by
the surface fluxes and radiative cooling. The sea surface
temperature (SST) distribution is given by a cosine-squared
function, with 303.15 K in the center and 299.15 K at the
periodic lateral boundaries. Radiative cooling is given
by the average profile of radiative tendency in the
radiative-convective simulation and by the relaxation
term toward the equilibrium value of potential temperature with the 20-day time scale.
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MONTHLY WEATHER REVIEW
A cloud-resolving model has been set up with
a 40 000-km horizontal scale and 24 km in the vertical.
Uniform resolution of 2 km in the horizontal direction
and 500 m in the vertical has been applied. The model
has been run with a time step of 15 s for over 320 days,
and the last 270 days have been analyzed. More detailed description of the modeling setup can be found
in SPMG, which discusses results from the cloudresolving simulation that provide the reference for SP
simulations.
In SP simulations, the large-scale domain spans
40 000 km with horizontal and vertical grid lengths of
48 km and 500 m, respectively. The large-scale time step
is 180 s. The cloud-resolving domain has horizontal and
vertical grid lengths of 2 km and 500 m, respectively, and
a small-scale time step of 15 s. The simulations are run
for 340 days, and the last 290 days are analyzed. Because
of the simulation length, no other SP setups [i.e., either
larger or smaller large-scale model grid length; cf.
section 3 of Grabowski (2001)] were considered. Simulations with various time accelerations and space compressions are compared. The horizontal domain of the
cloud-resolving model is equal to the large-scale horizontal grid length (i.e., px 5 1) or is reduced by a factor
of 2 ( px 5 2) or 3 (px 5 3). Also, for every large-scale
time step, the time integration in the cloud-resolving
domains is performed for the period either equal (pt 5
1) or 2 (pt 5 2), 3 ( pt 5 3), and 4 ( pt 5 4) times shorter
than the large-scale model time step. A simulation with
a given spatial compression ( px) and temporal acceleration ( pt) will be referred to as ‘‘SSTSPpxpt simulation.’’ For instance, a simulation with px 5 2 and pt 5 3
will be called ‘‘SSTSP23 simulation.’’ In total, 12 simulations are performed with different time accelerations
and space compressions. We will refer to them as ‘‘SSTSP
simulations.’’ SSTSP simulations are compared to the
benchmark case obtained with the cloud-resolving model
(CRM) and analyzed in SPMG, and referred to as the
‘‘CRM simulation’’ thereafter.
3. Results
SSTSP simulations reproduce the key characteristics
of the CRM simulation. In particular, large-scale overturning circulation is simulated in the large-scale domain, with the large-scale ascent over warm pool and
subsidence over cold SSTs. Similarly to the CRM simulation, variability across a wide range of scales is simulated. We start with a discussion of the mean state.
Subsequently, we present analysis of high- and lowfrequency variability, with the latter analyzed in more
detail. The emphasis is on comparing the SP and CRM
simulations (the latter one documented in detail in
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SPMG) and evaluating the impact of the spatial and
temporal scaling factors px and pt.
a. The mean Walker cell circulation
Figure 1 shows the time-averaged horizontal velocity
field for the CRM and SSTSP simulations (the former
already shown in Fig. 2a in SPMG) as well as the difference between them. The CRM large-scale circulation
features surface and midtropospheric mean horizontal
flows toward the highest SST in the center of the domain. The horizontal velocity maxima are around 10
and 5 m s21 at the surface and around 6-km altitude,
respectively. The upper-tropospheric outflow from
the center of the domain features maximum velocities
of over 20 m s21. SSTSP simulations exhibit similar
large-scale circulation, with low- and midlevel convergence accompanied by the upper-tropospheric divergence over warm SSTs, that is, with the first and
second baroclinic modes. The most apparent difference
between CRM and SSTSP simulations is the narrower
ascending region in the center of the domain in SSTSP
cases. Although the patterns and amplitudes of the
horizontal flow are similar in all simulations, the difference plots between CRM and SSTSP show significant deviations that seem to increase with the spatial
compression and temporal acceleration, with the SP
simulation without compression and acceleration (i.e.,
SSTSP11) being the closest to CRM as one might expect. Although not shown in the figure, the differences
depend primarily on the horizontal extent of the SP
domains (i.e., they increase with the increase of px), and
there seems to be no systematic impact of the temporal
acceleration (i.e., increasing the pt parameter).
Figures 2 and 3 document the impact of spatial
compression and time acceleration on the mean (i.e.,
horizontal and time averaged) profiles of the potential
temperature and water vapor mixing ratio. Figure 2
shows the difference between profiles from SSTSP with
various spatial compressions and CRM. Mean profiles
for the SSTSP11 simulation are close to CRM, and the
differences increase with the spatial compression. The
SSTSP31 simulation features up to 8-K colder upper
troposphere and up to 2 g kg21 lower moisture in the
lower troposphere when compared to CRM. The relative humidity profiles (not shown) agree relatively well
below 8 km for all simulations and differ significantly
above 10 km, with no obvious sensitivity to the spatial
compression. The differences between the temperature profiles are consistent with a heuristic argument
that reducing the horizontal extent of SP computational domains (i.e., increasing px) makes convective
overturning more difficult and leads to a colder upper
troposphere. The water vapor difference profiles can be
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FIG. 1. Time-averaged cross section of horizontal velocity (m s21) for the (top) cloud-resolving model and (left,
from top to bottom) SSTSP11, SSTSP22, and SSTSP33. Time-averaged cross section of horizontal velocity difference between (right, from top to bottom) SSTSP11, SSTSP22, and SSTSP33, and the cloud-resolving model.
explained by a narrower ascending region in the center
of the domain as illustrated in Fig. 1 and further quantified below. As shown in Fig. 3, time acceleration leads
to the mean temperature (moisture) profiles that are
warmer (more humid), but the effects are significantly
smaller than for the spatial compression, especially for
the moisture.
Figures 4 and 5 show spatial distributions of the difference between the SSTSP and CRM simulations for the
mean temperature and water vapor mixing ratio, respectively. The differences are averaged over days 50–
340. For the temperature, the patterns are dominated by
the differences in the mean profiles (cf. Fig. 2), with small
gradients between regions with high and low SST (i.e.,
mean ascent and mean subsidence). In the CRM simulation, the temperature field at a given level is homogenized by convectively generated gravity waves that
maintain a small horizontal temperature gradient. Such
a mechanism is also efficient in SP simulations, including
SSTSP, as documented by the relatively small horizontal
temperature gradients in Fig. 4. The water vapor field, on
the other hand, can only be homogenized by the physical
advection and the differences between SSTSP and CRM
simulations are larger, as shown in Fig. 5. The largest
differences (in the absolute sense) are near the center of
the domain, likely because of the different width of the
central ascending region and differences in the large-scale
circulation (cf. Fig. 1). The differences increase with the
increase of the spatial compression and temporal acceleration. The lower troposphere above 1 km is drier in
SSTSP than in the CRM, in both the ascent and subsidence regions, perhaps with the exception of the
narrow zone over the coldest SSTs. The level of maximum difference outside the central region at heights
between 2 and 3 km corresponds to the low-level cloud
tops (see below). The upper troposphere is drier at the
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FIG. 2. (top) Mean profile of (left) potential temperature (K) and (right) water vapor mixing ratio (kg kg21) for
the cloud-resolving model. (bottom) Difference in the mean profiles of (left) potential temperature (K) and (right)
water vapor mixing ratio (kg kg21) between SSTPS11 (red line), SSTSP21 (green line), and SSTSP31 (black line),
and the cloud-resolving model.
warm pool edges, likely because of the narrower region
of deep convection in the SSTSP simulations.
Figures 6 and 7 show time-averaged mean fields and
profiles of the cloud condensate mixing ratio, respectively.
Figure 6 shows that shallow convection occurs over the
entire domain, while deep convection is confined to the
warm pool. The region with deep convection narrows
when the spatial compression and temporal acceleration
increase. There are also systematic changes of the mean
cloud condensate profiles as documented in Fig. 7. The
figure documents the classical trimodal characteristics
of the tropical moist convection: shallow, congestus, and
deep (cf. Johnson et al. 1999), with the lower-tropospheric
maximum associated with shallow convective clouds, and
mid- and upper-tropospheric maxima marking detrainment levels from congestus and deep convection, respectively. Temporal acceleration results in a significant
shift of the profiles toward higher values (a factor of approximately 2 between Figs. 7a and 7d). Spatial compression for a given temporal acceleration has a relatively
smaller effect, with a systematic decrease of cloud condensate above 5 km.
Figure 8 shows mass flux profiles for CRM and SSTSP
simulations with various spatial compressions and
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571
FIG. 3. Difference in the mean profiles of (top) potential temperature (K) and (bottom) water vapor mixing ratio (kg kg21) between
(left) SSTSP11, (middle) SSTSP21, and (right) SSTSP31, and SSTSP simulations with the same spatial acceleration px and temporal
acceleration pt 5 2, 3, and 4 (dotted, dashed, and dashed–dotted lines, respectively).
temporal accelerations. Since these profiles are derived
by averaging the cloud-model data, they represent the
impact of the SSTSP methodology on convective transport. The SP simulation with neither spatial compression
nor temporal acceleration (i.e., SSTSP11) gives the mean
mass flux close to the one from the CRM simulation.
Spatial compression (i.e., SSTSP31) leads to significantly
modified vertical velocity distribution (see Fig. 9) and thus
reduced mass flux, arguably because of the impact of a reduced extent of the cloud-model computational domain on
the convective transport as argued at the end of section 2a.
In contrast, temporal acceleration (i.e., SSTSP13) leads to
a significant increase of the mass flux, arguably because of
the increase of the large-scale forcing [cf. (7) and (13)].
Combining spatial compression and temporal acceleration
(i.e., SSTSP33) results in the convective mass flux in between the ones obtained from the simulations with either
spatial compression or temporal acceleration.
The differences in the convective mass flux affect the
mean (domain and time averaged) profiles of the precipitation water mixing ratio as shown in Fig. 10. The
simple microphysics parameterization used in the simulations assumes precipitation to be in the form of snow
(rain) in the upper (lower) troposphere, with snow
sedimenting with significantly smaller vertical velocity.
This explains the difference between lower- and uppertropospheric values of each profile. However, the magnitude of the profiles [i.e., the largest (smallest) for SSTSP13
(SSTSP31)] is in direct response of the convective mass
flux shown in Fig. 8. The difference between precipitation water profiles might have a significant impact
on model results once an interactive radiation scheme is
used in place of a prescribed radiative cooling applied in
current simulations.
The SSTSP framework significantly modifies the spatial distribution of convection and related statistics. The
differences in cloudiness are associated with different
spatial distributions of the time-averaged precipitable
water content, cloud-top temperature, and precipitation
rate, as shown in Fig. 11, respectively, with their mean
values given in Table 1. As the figures document, SSTSP
simulations are characterized by significantly narrower
distributions of all the quantities. In the CRM simulation,
the central 10 000 km is characterized by the mean cloud
top temperature around 288 K, precipitable water around
75 kg m22, and surface precipitation around 0.45 mm h21.
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MONTHLY WEATHER REVIEW
FIG. 4. (top) Time-averaged cross section of potential temperature (K) for the cloud-resolving model. Time-averaged cross section
of the potential temperature difference (K) between (second, third,
and fourth rows) SSTSP11, SSTSP22, and SSTSP33, and the cloudresolving model.
All distributions are relatively flat and feature steep
gradients at the edges of the warm pool with the mean
precipitation dropping below 0.1 mm h21 and mean cloudtop temperature increasing to around 300 K. SSSTP simulations, on the other hand, are characterized by narrow
distributions, with peaks at the center and steep gradients of the mean cloud-top temperature and precipitation. These differences also occur in SSTSP11, that
is, the SP simulation with neither time acceleration nor
spatial compression, and thus are a general feature of
the SP simulation.
Because of the complicated impact of the time acceleration on diabatic processes, an intrinsic feature of the
SSTSP framework, it is impossible to rescale the cloudtop temperature between CRM and SSTSP simulations.
Increasing temporal acceleration leads to more intense
convective activity (cf. Fig. 8), increased cloudiness and
precipitable water, and a decreased mean cloud-top
temperature (cf. Table 1). These aspects of temporal acceleration have been pointed out by Pauluis et al. (2006)
and they seem to be related to the way microphysical
VOLUME 143
FIG. 5. (top) Time-averaged cross section of water vapor mixing
ratio (kg kg21) for the cloud-resolving model. Time-averaged cross
section of the water vapor mixing ratio difference (kg kg21) between (second, third, and fourth rows) SSTSP11, SSTSP22, and
SSTSP33, and the cloud-resolving model.
processes (in particular fallout of rain) are handled. No
acceleration of microphysical processes is applied here,
potentially impacting the balance between processes
responsible for moistening and drying the troposphere.
In summary, the SSTSP framework appears to simulate the 2D Walker circulation qualitatively well. In
particular, the mean large-scale flow consists of a deep
overturning circulation (first baroclinic mode) and
a midtropospheric jet (second baroclinic mode). Deep
convection occurs primarily over the warm pool, and
subsidence regions are dominated by shallow convection. However, detailed comparison reveals systematic
differences in the model mean state. These differences are mainly artifacts of the original implementation of SP with additional drawbacks introduced along
with SSTSP framework. The artificial scale separation
between large-scale and small-scale models and the
periodicity of small-scale models impose significant
limitations on the flow field in the small-scale domain
(e.g., vanishing mean mass flux) and subsequently on
the simulated convection and its organization. Temporal acceleration of large-scale forcing is associated,
in turn, with enhancement of convection of which nonlinear nature appears to be an obstacle for proper scaling
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SLAWINSKA ET AL.
FIG. 6. (top) Time-averaged cross section of the cloud water
mixing ratio (kg kg21) for the cloud-resolving model. Time-averaged
cross section of the cloud water mixing ratio difference (kg kg21)
between (second, third, and fourth rows) SSTSP11, SSTSP22, and
SSTSP33, and the cloud-resolving model.
of cloudy properties, impacting significantly, for example, the radiative calculation. Moreover, the convective
feedback to the large scale cannot be easily modified,
which is associated with a negative impact on mean
large-scale conditions.
b. Transients
SPMG document several transient features occurring
in CRM simulation. The large-scale flow is characterized
by low-frequency variability featuring 20-day oscillations with alternating periods of strong and weak overturning circulation. The strong circulation phase is
associated with intense convection and expansion of the
large-scale convergence region over the warmest SSTs.
The weak circulation phase, on the other hand, features
reduced convective activity and a narrower convergence
region. The expansion (compression) of the convergence region coincides with synoptic-scale convective
activity propagating from (to) the center of the domain
with the average speed between 5 and 10 m s21.
Here, we investigate if the SSTSP framework is capable of capturing these oscillations. Figure 12 shows
Hovmoeller diagrams of cloud-top temperature for
SSTSP11 and SSTSP33 simulations. The figure shows
that the variability in SP simulations is of a similar
character to that in the CRM simulation. In particular,
573
the zigzag pattern formed by very cold tops of convective
cloud systems propagating toward and then away from
the convergence region apparent in the CRM simulation
can also be noticed in the SP model. However, the coherency decreases with the increase of spatial compression and temporal acceleration. This is consistent with the
fact that coherent propagation of convective-scale features across the SP model grid is more difficult than in the
CRM model because the cloud-scale models only communicate through the large-scale model dynamics. Another feature apparent in Fig. 12 is that convection seems
to be more localized in the center of the domain as already documented in Figs. 11 and 12.
An inability to simulate an organized convective system in climate models has been documented before
(Pritchard and Somerville 2009a,b). Just recently propagation of convective systems has been observed in SPCAM3.5 and SP-CAM5 by Pritchard el al. (2011) and
Kooperman et al. (2013). In their study, they reproduce
nocturnal convection crossing the Great Plains east of the
Rocky Mountains in a specific implementation of SPCAM. In particular, they find it sensitive to such factors as
microphysical parameterization, horizontal resolution of
a cloud-resolving model, and the choice of the dynamical
core, with no detailed explanation of these dependencies.
Although they attribute the propagation of mesoscale
convective systems to a large-scale first-baroclinic wave
driven by convective heating, they do not exclude other
mechanisms (e.g., large-scale advection of water vapor).
As such, an understanding of propagation of convective
systems remains elusive and more studies are needed to
overcome that. Here we apply SSTSP and systematically
modify the coupling of the large-scale and the cloudresolving model, finding the propagation of convective
systems impacted negatively both by spatial compression
and temporal acceleration. We hope to report in the future on the details of physical mechanisms associated
with these dependencies.
c. Low-frequency variability
Low-frequency variability in the CRM simulation has
been analyzed in detail in SPMG. There, we apply the
empirical orthogonal function (EOF; von Storch and
Zwiers 1999) analysis and develop an index of the lowfrequency variability. Subsequently, we construct a composite of low-frequency variability with a lag-regression
analysis applied to the index. We analyze reconstructed
fields of different dynamical variables and describe the
low-frequency oscillation. We find the low-frequency
variability of a 20-day period, triggered by anomalously
intense deep convection over the warm pool. This, in
turn, is the consequence of large-scale horizontal advection of anomalously moist air from the subsidence region
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FIG. 7. Mean profiles of the cloud water mixing ratio (kg kg21) for the cloud-resolving model (red solid) and
SSTSP simulations with spatial compression px 5 1, 2, and 3 (solid, dashed, dotted–dashed black, respectively) and
(from top left to bottom right) temporal acceleration pt 5 1, 2, 3, and 4, respectively.
after the period of moisture buildup through anomalously
intense shallow convection.
Here, we investigate if the low-frequency variability is captured by applying the same methodology
as for the CRM simulation in SPMG. First, we perform EOF analysis for the last 290 days of large-scale
surface wind data with 1-h temporal resolution.
Subsequently, for every SSTSP simulation, we analyze the low-frequency variability by applying the
principal component of the leading EOF (see section 4 in SPMG). Table 2 presents the main characteristics of the leading EOF for various SSTSP
simulations and for the CRM simulations from SPMG.
All SSTSP simulations exhibit a dominant mode of lowfrequency variability corresponding to a strengthening/
weakening of the low-level flow as identified previously for the CRM simulation in SPMG. The power
spectrum peaks for the period in between 23 and 26
days and compares reasonably well with the 20-day
period of the CRM simulation. It thus appears that the
SSTSP framework captures the variability corresponding to the intraseasonal band that, in turn, is responsible for a significant percent of the total variance
as in the CRM simulation. Overall, SSTSP simulations
with a larger spatial compression or temporal acceleration tend to exhibit lower total variance. SSTSP23 and
SSTSP24 simulations feature the closest variance to the
CRM simulation.
To characterize low-frequency oscillation in more
detail, composites of low-frequency variability are
FIG. 8. Mean cloudy (qc . 0.000 01 kg kg21) mass flux profiles for
SSTSP simulations with spatial compression 1 and 3 (red and black
line, respectively) and temporal acceleration 1 and 3 (solid and
dashed line, respectively).
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SLAWINSKA ET AL.
575
FIG. 9. Logarithm of frequency distribution of vertical velocity for cloudy (qc . 0.000 01 kg kg21) fields for (top left)
SSTSP11, (top right) SSTSP13, (bottom left) SSTSP31, and (bottom right) SSTSP33.
constructed by a lag regression of a given variable
anomaly f(x, t) against the first EOF as follows:
1
h f i(x, t) 5
s
ð
PC1 (t)f (x, t 1 t) dt ,
where s stands for the standard deviation of the first
EOF, PC1. Finally, we obtain the composite by adding
the mean value of the given field f (x, t).
All SSTSP simulations reproduce phases of the lowfrequency oscillation, with the exemplary composite of
the horizontal velocity for SSTSP22 simulation shown in
Figs. 13 and 14. Figure 13 can be compared to Fig. 8 in
SPMG, whereas Fig. 14 can be compared to Figs. 9a to
12a in SPMG. As in SPMG, the low-frequency oscillation consists of four phases: suppressed, strengthening,
active, and decaying. The mechanisms behind low-frequency
oscillation are robust and are reproduced in all SSTSP
simulations. As in the CRM simulation, large-scale advection of moisture is correlated with oscillations of convective activity and large-scale circulation. The suppressed
phase is characterized by weak large-scale overturning
circulation and decreasing deep convective activity in the
central part of the domain. This, in turn, is associated with
a drier troposphere due to an anomalously strong midtropospheric advection of dry air from the subsidence region and an anomalously weak advection of the moist
surface air to the central part of the domain. At the same
time, anomalously weak subsidence allows for moisture
buildup over the subsidence region, as shallow convective activity intensifies. Circulation strengthens as lowtropospheric anomalously moist air is advected into the
central part of the domain and deep convection intensifies.
As deep convective activity reaches its peak, the troposphere warms and dries as a result of the latent heat release
and intense precipitation. The decaying phase follows when
the central region dries out because of the intense precipitation, and it is accompanied by a strong midtropospheric jet bringing dry air from the subsidence region and
FIG. 10. As in Fig. 8, but for the domain-averaged rainwater mixing
ratio (kg kg21).
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TABLE 1. Mean precipitable water content (Prw; kg m22), cloudtop temperature (Cltop; K), and precipitation (Precip; mm h21).
FIG. 11. Time-averaged horizontal distribution of (top) precipitable water (kg m22), (middle) cloud-top temperature (K), and
(bottom) precipitation (mm h21) for SSTSP11 (solid red line),
SSTSP22 (dotted green line), SSTSP33 (dashed black line), and the
cloud-resolving model (solid magenta line). Mean values for all
simulations are given in Table 1.
advection of anomalously dry low-tropospheric air as the
shallow convection weakens due to strong subsidence.
4. Discussion and conclusions
The primary purpose of this paper is to evaluate the
sparse space–time superparameterization (SSTSP) introduced in Xing et al. (2009). SSTSP extends the original
superparameterization (SP) approach, where convective
processes are simulated explicitly by a cloud-resolving
pxpt
Prw
Cltop
Precip
11
12
13
14
21
22
23
24
31
32
33
34
CRM
53.8
54.2
54.6
54.8
51.0
51.1
51.4
51.1
47.0
47.2
47.3
47.6
55.9
297.3
294.4
291.9
290.0
297.3
294.5
292.4
290.5
297.5
295.2
293.0
291.3
296.5
0.128
0.130
0.133
0.133
0.122
0.123
0.125
0.125
0.108
0.110
0.111
0.111
0.157
model embedded in every large-scale model column.
The motivation behind SSTSP comes from the quest
for computationally affordable and statistically accurate simulations of large-scale circulation that crucially
depend on convective activity. SSTSP addresses this
issue by significantly reducing cloud-resolving calculations and at the same time assuring its statistically accurate small-scale (convective) feedback. The feedback
is obtained by rescaling statistics from cloud-resolving
calculations over time and horizontal domain spans that
are reduced relative to the large-scale time and space
resolutions.
Xing et al. (2009) performed the initial tests of the
SSTSP methodology. They conducted idealized simulations of a squall line within 1000-km horizontal domain for a 6-h period. It was shown that SSTSP captures
propagation of a squall line across the domain, with the
propagation speed controlled by the prescribed shear.
Here, we evaluate the SSTSP framework for an idealized Walker cell setup. This is a more complex case than
in Xing et al. (2009) because it features a wider range of
spatiotemporal scales, up to planetary scales and time
periods up to several tens of days. In contrast to Xing
et al. (2009), larger-scale flow can evolve in response to
feedbacks from moist convection, and in turn it can affect
the subsequent convective development. We evaluate the
performance of the SSTSP algorithm by comparing solutions to those obtained applying the cloud-resolving
model (CRM) described in SPMG.
We find that SSTSP is capable of reproducing key
characteristics of the cloud-resolving Walker cell simulation. In agreement with CRM results, the mean state
features the first and second baroclinic modes with deep
convection over high SST organized into propagating
systems. The properties of these convective systems
(e.g., structure, propagation) are affected by the horizontal extent of SP cloud-resolving domains. This is an
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577
FIG. 12. Hovmoeller diagrams of the cloud-top temperature for (left) SSTSP11 and (right) SSTSP33 simulations.
artifact of the periodicity of embedded cloud-resolving
models that cannot be avoided. SSTSP captures intraseasonal variability predicted by the CRM model that
consists of four distinctive stages—suppressed, intensification, active, and weakening—along with the mechanisms driving them. Differences in convective evolution
and propagation result in some differences between
SSTSP and CRM simulations, such as in the mean
sounding, spatial distribution of the cloudiness, and surface precipitation, or in the spatiotemporal characteristics
of the low-frequency oscillation. Differences in the mean
cloudiness (cf. Fig. 9) will most likely be accentuated
when an interactive radiation transfer scheme is used, an
aspect not addressed in the current study.
Numerical simulations discussed here can be put into
the context of those discussed in Grabowski (2001,
hereafter G01). Section 3 of G01 presented SP simulations of a 2D flow driven by large-scale SST gradients,
although of a significantly smaller horizontal extent
TABLE 2. Eigenvalue periods of the first EOF for the surface
wind time series and standard deviation of its principal component.
Simulations with SSTSP algorithm (with px and pt as given in the
first column). CRM results are included in the bottom row.
pxpt
Eigenvalue
Period (days)
Std dev
11
12
13
14
21
22
23
24
31
32
33
34
CRM
0.51
0.53
0.54
0.42
0.41
0.29
0.39
0.33
0.29
0.22
0.18
0.18
0.36
24
26
24
24
26
24
26
26
24
23
23
26
20
157
154
159
133
117
85
108
95
86
71
62
63
88
FIG. 13. Hovmoeller diagrams of (top) lag-regressed surface wind
(m s21), (middle) surface precipitation rate (mm h21), and (bottom)
precipitable water content (kg m22) for the SSTSP22 simulation.
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FIG. 14. Lag-regressed structure of the horizontal velocity anomaly for the (a) suppressed phase, (b) strengthening
phase, (c) active phase, and (d) decaying phase for the SSTSP22 simulation.
(a computational domain of 4000 km in G01 rather than
40 000 km here and in SPMG). The SP simulations in
section 3 of G01 were compared to CRM simulations
discussed in Grabowski et al. (2000). G01’s SP simulations used various horizontal grid lengths of the largescale model (from 20 to 500 km; referred to as P20 and
P500, respectively) with the horizontal extent of the
embedded CRM model matching the large-scale model
grid length. G01’s results documented a significant impact of the specific setup of the SP model configuration
(i.e., from P20 to P500), especially for the mesoscale
convective organization (cf. Fig. 8 therein). In contrast,
the SP simulations presented here feature just a single
48-km horizontal grid length of the large-scale model
and explore the impact of SSTSP methodology. Such
a grid length can be argued to follow a recommendation
of Grabowski (2006a) who suggested that the SP
approach is better suited for large-scale models with
horizontal grid lengths in the mesoscale range (i.e., a few
tens of kilometers). This is because, in the mesoscale
grid length case, the embedded SP models represent the
effects of small-scale convective motions only, and the
convective mesoscale organization (e.g., into squall
lines) can be simulated by the large-scale model. Both
convective and mesoscale circulations have to be represented by the SP model when the large-scale model
grid length is hundreds of kilometers, as in typical global
climate applications (e.g., Khairoutdinov et al. 2005;
DeMott et al. 2007, among others). Section 4 of G01 applies the SP methodology to the problem of large-scale
convective organization on an idealized constant-SST
(‘‘tropics everywhere’’) aquaplanet. Although not emphasized there, the SP aquaplanet simulations [as well
as subsequent studies, e.g., Grabowski (2006b)] already
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SLAWINSKA ET AL.
apply the spatial compression methodology because of the
disparity between the large-scale model grid length and
the horizontal domain size of the embedded CRM model.
The obvious main drawback of the SP approach is that
every cloud-resolving model is independent of each
other and communicates solely by the large-scale model
dynamics (i.e., through the large-scale forcings). The key
point is that cloud systems cannot propagate directly
from one large-scale grid box to the other, but they remain locked in a single large-scale grid box because of
the periodic lateral boundary conditions. As discussed
by Jung and Arakawa (2005), periodic lateral boundary
conditions require the mean mass flux to vanish. As
a result, updrafts get weaker as the horizontal extent of
the cloud-resolving domain decreases. Large-scale
thermodynamical fields are also modified, for instance,
the lower (upper) troposphere becomes moister (drier).
This key drawback of the SP (and thus SSTSP) approach
was also noted in other studies (e.g., G01) and it is evident in our simulations. To overcome these limitations,
Jung and Arakawa (2005) suggest an alternative approach where periodic boundary conditions are abandoned and the adjacent cloud-resolving models are
linked allowing for the direct propagation of small-scale
perturbations from one large-scale model grid box to another. Such an approach is relatively straightforward in
a 2D large-scale model framework, but it requires more
complicated parallel-processing methods as opposed to the
‘‘embarrassingly parallel’’ logic of the original SP. Perhaps
more importantly, this simple idea leads to a significantly
more complex methodology when implemented into a 3D
large-scale model (cf. Jung and Arakawa 2010, 2014).
Considering these factors, we feel that the traditional SP/
SSTSP methodology can still serve as a valuable technique
in large-scale models featuring mesoscale horizontal grid
lengths and variable orientation of SP cloud-resolving
models (cf. Grabowski 2004). We hope to report on such
numerical experiments in forthcoming publications.
Acknowledgments. J. Slawinska acknowledges funding from CMG Grant DMS-1025468 as well as support
from the Center for Prototype Climate Modeling at
NYU Abu Dhabi. This research was carried out on the
high performance computing resources at New York
University Abu Dhabi.
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